Chapter 10: Problem 6

About \(30 \%\) of the \(1,360 \mathrm{~W} / \mathrm{m}^{2}\) solar power arriving at Earth is immediately bounced back without a trace. Of the part that remains, when distributed/averaged around the whole sphere, \(^{34}\) what is the average energy deposition rate per square meter into the earth system?

Short Answer

Expert verified

Answer: The average energy deposition rate per square meter into the Earth's system is approximately 237.6 W/m^2.

Step by step solution

Calculate the total solar power arriving at Earth

To do this, we will use the given solar power value, 1360 W/m^2. Total solar power = 1360 W/m^2

Calculate the part of solar power not bounced back

About 30% of the total solar power is immediately bounced back into space. Hence, we need to find the 70% of it that is actually absorbed by Earth. Remaining solar power = Total solar power × (1 - 0.30) = 1360 W/m^2 × 0.70 = 952 W/m^2

Find the Earth's surface area

Earth can be approximated as a sphere with a radius of 6,371,000 m. We will use the formula for the surface area of a sphere, which is: Surface area = 4 * π * radius^2 Earth's surface area = 4 * π * (6,371,000 m)^2 ≈ 5.1 × 10^14 m^2

Calculate the solar energy absorption rate

To find the total solar energy absorbed by the Earth, we will multiply the remaining solar power (from step 2) by the incoming area of solar power, which is the cross-sectional area of Earth. Cross-sectional area of Earth = π * radius^2 ≈ 1.27 × 10^14 m^2 Total solar energy absorbed by the Earth = Remaining solar power × Cross-sectional area of Earth ≈ 952 W/m^2 × 1.27 × 10^14 m^2 ≈ 1.21 × 10^17 W

Calculate the average energy deposition rate per square meter

To find the average energy deposition rate per square meter into the Earth's system, we will divide the total solar energy absorbed (from step 4) by the Earth's surface area (from step 3). Average energy deposition rate = Total solar energy absorbed / Earth's surface area ≈ 1.21 × 10^17 W / 5.1 × 10^14 m^2 ≈ 237.6 W/m^2 The average energy deposition rate per square meter into the Earth's system is approximately 237.6 W/m^2.

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Short Answer

Step by step solution

Calculate the total solar power arriving at Earth

Calculate the part of solar power not bounced back

Find the Earth's surface area

Calculate the solar energy absorption rate

Calculate the average energy deposition rate per square meter

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